Problem 66 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the discriminant to determine the number of real solutions of the equation. $$ 2 p^{2}+5 p+6=0 $$

Step-by-Step Solution

Answer

The given quadratic equation is \(2p^{2} + 5p + 6 = 0\), where \(a = 2\), \(b = 5\), and \(c = 6\). Using the discriminant formula, \(D = b^2 - 4ac\), we calculate the discriminant as \(D = (5)^2 - 4(2)(6) = 25 - 48 = -23\). Since \(D < 0\), the equation has no real solutions and two complex solutions.

Step 1: Identify the coefficients

From the given equation \(2p^{2} + 5p + 6 = 0\), we can see that \(a = 2\), \(b = 5\), and \(c = 6\). Step 2: Use the formula to find the discriminant

Step 2: Calculate the discriminant

The formula for the discriminant is given by: \(D = b^2 - 4ac\). Substitute the values \(a = 2\), \(b = 5\), and \(c = 6\) into the formula: \[D = (5)^2 - 4(2)(6)\] Step 3: Evaluate the discriminant

Step 3: Evaluate the discriminant

By calculating the values, we get: \[D = 25 - 48\] \[D = -23\] Step 4: Determine the number of real solutions

Step 4: Determine the number of real solutions

Based on the value of the discriminant, we can determine the number of real solutions: - If \(D > 0\), then the equation has two distinct real solutions. - If \(D = 0\), then the equation has one real solution (a double root). - If \(D < 0\), then the equation has no real solutions (two complex solutions). Since our discriminant value is \(D = -23\), which is less than zero, our equation has no real solutions, and we can conclude that it has two complex solutions.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra, characterized by the presence of a variable raised to the power of two. They take the general form: \\[ax^2 + bx + c = 0\] where:

• \(a\), \(b\), and \(c\) are constants.
• \(x\) represents the variable.

The solutions to quadratic equations represent the points at which the parabola intersects the x-axis, also known as the roots of the equation. The main methods for solving quadratic equations include:

• The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Factoring, when possible
• Completing the square

Each method has its own usage scenarios, but the Quadratic Formula is universally applicable, as it works for all types of quadratic equations, regardless of whether the roots are real or complex.

Real Solutions

In the realm of quadratic equations, real solutions are the values of \(x\) that satisfy the equation and lie on the real number line. Determining the number of real solutions involves finding the value of the discriminant, \(D\), calculated as: \\[b^2 - 4ac\]
The discriminant tells us about the nature of the roots:

• If \(D > 0\), there are two distinct real solutions. This means the parabola crosses the x-axis at two different points.
• If \(D = 0\), there is exactly one real solution, which is known as a repeated root or double root. In this scenario, the parabola touches the x-axis at exactly one point.
• If \(D < 0\), there are no real solutions. This indicates that the parabola does not intersect the x-axis at any point.

Real solutions are critical in many mathematical applications, such as physics and engineering, where identifying real-world intersections is vital.

Complex Solutions

When a quadratic equation does not have real solutions, it instead possesses complex solutions. These emerge when the discriminant \(D\) is less than zero, suggesting that the parabola does not intersect the x-axis. Complex solutions take the form of complex numbers, typically represented as: \\[a + bi\] where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).
For quadratic equations, complex solutions always come in conjugate pairs, such as \(a + bi\) and \(a - bi\). Despite the lack of real intersections with the x-axis, complex solutions hold significant value in theoretical mathematics and complex signal processing.
Visualizing these solutions can be insightful in understanding the concept of imaginary numbers, further broadening our comprehension of quadratic equations beyond the real number line.